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\begin{document}

\title{A statistical approach for spatial mapping and temporal forecasts of
volcanic eruptions using monitoring data}



\author[1]{Abani Patra}%
\author[2]{Andrea Bevilacqua}%
\author[3]{Eric Bruce Pitman}%
\author[3]{Marcus Bursik}%
\author[4]{Barry Voight}%
\author[2]{Augusto Neri}%
\author[2]{Giovanni Macedonio}%
\author[5]{Franco Flandoli}%
\author[2]{Prospero De Martino}%
\author[2]{Flora Giudicepietro}%
\author[6]{Stefano Vitale}%
\affil[1]{Tufts University}%
\affil[2]{Istituto Nazionale di Geofisica e Vulcanologia}%
\affil[3]{University at Buffalo}%
\affil[4]{Pennsylvania State University}%
\affil[5]{Scuola Normale Superiore di Pisa}%
\affil[6]{Università di Napoli Federico II}%


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  \date{May 25, 2024}


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\begin{abstract}
We present two models using monitoring data in the production of
volcanic eruption forecasts. The first model enhances the
well-established failure forecast method introducing an SDE in its
formulation. In particular, we developed new method for performing
short-term eruption timing probability forecasts, when the eruption
onset is well represented by a model of a significant rupture of
materials. The method enhances the well-known failure forecast method
equation. We allow random excursions from the classical solutions. This
provides probabilistic forecasts instead of deterministic predictions,
giving the user critical insight into a range of failure or eruption
dates. Using the new method, we describe an assessment of failure time
on present-day unrest signals at Campi Flegrei caldera (Italy) using
either seismic count and ground deformation data. The new formulation
enables the estimation on decade-long time windows of data, locally
including the effects of variable dynamics. The second model establishes
a simple method to update prior vent opening spatial maps. The prior
reproduces the two-dimensional distribution of past vent distribution
with a Gaussian Field. The likelihood relies on a one-dimensional
variable characterizing the chance of material failure locally, based,
for instance, on the horizontal ground deformation. In other terms, we
introduce a new framework for performing short-term eruption spatial
forecasts by assimilating monitoring signals into a prior
(``background'') vent opening map. To describe the new approach, first
we summarize the uncertainty affecting a vent opening map pdf of Campi
Flegrei by defining an appropriate Gaussian random field that replicates
it. Then we define a new interpolation method based on multiple points
of central symmetry, and we apply it on discrete GPS data. Finally, we
describe an application of the Bayes' theorem that combines the prior
vent opening map and the data-based likelihood product-wise. We provide
examples based on either seismic count and interpolated ground
deformation data collected in the Campi Flegrei volcanic area.%
\end{abstract}%



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